Animating Impossible Objects Peter Kovesi and Chih Khoh

Animating Impossible Objects Peter Kovesi and Chih Khoh School of Computer Science & Software Engineering The University of Western Australia

An impossible figure is a two-dimensional image that is interpreted to give the impression of some three-dimensional object that cannot exist.

vision graphics 3 D model image

! vision 3 D model ? graphics image

Impossible shading. . .

Uccello: The Battle of San Romano ~1430

Uccello: The Hunt ~1460

Uccello Drawing of a Chalice

False Perspective, William Hogarth (1753)

Giovanni Battista Piranesi 14 th Prison (1760)

Swedish artist Oscar Reutesvard was the first to intentionally construct impossible figures. He devised this version of the impossible tri-bar in 1934

Oscar Reutesvard

In 1958 Penrose independently devised the impossible tri-bar and published a paper (with his dad) in the British Journal of Psychology. Correspondence between Penrose and Escher resulted in this image Waterfall (1961)

Penrose also devised the impossible staircase Ascending Descending (1960)

Why are Objects Impossible? Line Labeling Inconsistency + ^ ^^ Convex edge. Concave edge. Occluding edge (surface to the right). Apparent contour (surface to the right). Shigeo Fukuda (Huffman: Impossible Objects as Nonsense Sentences, 1971)

But some impossible objects can be labeled consistently…

The Aspect Graph (Koenderink and van Doorn 1979) Nodes: Generic views, or aspects of an object. Edges: Possible transitions between aspects. Aspect graph of a tetrahedron

Aspect Graph of a Cube An impossible object can result from the simultaneous presentation of two distant aspects of an object.

“An impossible figure is a two-dimensional image that is interpreted to give the impression of some three-dimensional object that cannot exist. ” But some impossible 3 D objects are possible…

Impossible triangle by Mathieu Hamaekers

A 3 D model must be handcrafted to suit the viewpoint. A computer model has an advantage in that it can be continuously adjusted to suit the viewpoint …

Constructing Impossible Figures via Complementary Halves An impossible rectangle and its two halves, each of which are globally consistent

One complementary half can be obtained from the other via reflections across two orthogonal axes

An impossible rectangle can also be created by reversing the visibility of the faces on one half of a possible rectangle

The Necker Cube and its two interpretations

Donald Simanek’s Ambiguous Ring

The Impossible Stall: The basis of Escher’s Belvedere

Model of Belvedere by Shigeo Fukuda

Model of Waterfall by Shigeo Fukuda

The Crazy Crate

Mathieu Hamaekers and his model of an impossible crate

Rotating the Impossible Rectangle

Animation Requires Continuous Modification of the 3 D Model Failure to adjust thickness during rotation produces halves that cannot be joined Note how the bars of this crazy crate must be non-square to allow joining

Algorithm 1. 2. 3. 4. 5. 6. 7. Construct 3 D model of one half of the object (origin at the centre point of join). Orient it to the desired view. Project into the image plane (orthographic projection). Calculate projected widths of surfaces to be joined. Rescale widths of corresponding surfaces on the 3 D model to allow joining in 2 D. Construct second half by negating X and Y coordinates (Z values unchanged). Add lines to the 2 D image to ‘fix’ the join as necessary.

Challenges… Impossible stereo/autostereograms. Impossible shading/lighting. Impossible motion. Non-even symmetry Line labeling inconsistency Model by Shigeo Fukuda

A computer model of Esher’s “High Low” by Sascha Ledinsky rendered in POV-ray.